The regularity theory for degenerate Kolmogorov equations relies on a suitable non-Euclidean geometry on the domain of the solutions. All the regularity results need to be stated in terms of this geometric structure. In particular, the degeneracy of the Kolmogorov equations yields a non isotropy in the Taylor expansion of functions that have a prescribed degree of regularity. I present some Schauder estimates for classical solutions to Kolmogorov equations in non divergence form with Dini-continuous coefficients. One of the main achievement of the article is the Taylor expansion under minimal regularity of solutions. I will also discuss a recent result concerning the Taylor expansion for a fractional kinetic equation. The Schauder estimate has been obtained in a joint work in collaboration with A. Rebucci and B. Stroffolini.

We will present a new approach to the problem of proving hydrodynamic limits from microscopic stochastic particle systems, namely the zero-range, the simple exclusion (jump) and the Ginzburg-Landau (diffusion) processes, to macroscopic partial differential equations. The qualitative behavior of the particle densities of such processes moving on a lattice according to a Markovian law, is well-known. . So far there are two major methods that work for a wide variety of models which are (i) due to Guo-Papanicolaou-Varadhan and (ii) the relative entropy method of H.-T Yau. Our method is simplified and unified that gives an explicit, uniform in time, rate of convergence to the limit PDE, under diffusive scaling (joint work with Daniel Marahrens and Clément Mouhot).

We consider two classical variational problems in the (not so classical) setting of conical sectors, and we discuss the characterization of their critical points. In this respect we present a Serrin-type and an Aleksandrov-type result in the context of convex cones. We focus on the role of the convexity of the cone and of the gluing between the cone and the relative boundary of the domain. We also show a rigidity result for constant mean curvature surfaces in starshaped sectors of non-convex cones, and its link with relative isoperimetric inequalities. This is a joint project with F. Pacella.

Myxobacteria are rod-shaped, social bacteria that are able to move on flat surfaces by ’gliding’ and form a fascinating example of how simple cell-cell interaction rules, including alignment and reversal of individuals, can lead to emergent, collective behaviour. In this talk a new kinetic model of Boltzmann-type for such colonies of myxobacteria will be introduced and investigated. For the spatially homogeneous case an existence and uniqueness result will be shown, as well as exponential decay to an equilibrium for the Maxwellian collision operator. Further, model extensions and their analysis will be addressed and numerical simulations will be shown.

In this talk, we will see that the set of singular times for weak solutions of the space homogeneous Landau equation with Coulomb potential constructed by C. Villani (1998) has Hausdorff dimension at most 1/2. This is a joint work with François Golse, Maria Pia Gualdani and Alexis Vasseur.

We will discuss some recent developments on the regularity theory for viscous Hamilton-Jacobi equations with superlinear first-order terms and unbounded right-hand side in Lebesgue spaces. In particular, we will address the so-called problem of maximal L^p-regularity both for stationary and time-dependent problems, and its consequences on the regularity theory for PDE systems arising in the theory of Mean Field Games. The results answer positively to a conjecture raised by P.-L. Lions. The talk will be based on joint works with M. Cirant (Padova).

For cross-diffusion systems possessing an entropy (i.e. a Lyapunov functional) we study nonlocal versions and exhibit sufficient conditions to ensure that the nonlocal version inherits the entropy structure. These nonlocal systems can be understood as population models per se or as approximation of the classical ones. With the preserved entropy, we can rigorously link the approximating nonlocal version to the classical local system. From a modelling perspective this gives a way to prove a derivation of the model and from a PDE perspective this provides a regularisation scheme to prove the existence of solutions. A guiding example is the SKT model and in this context we answer positively to a question raised by Fontbona and Méléard and thus provide a full derivation.

The aim of the talk is to introduce a new class of variational inequalities in which every operator is defined in a tensor Hilbert space (see [1]), called tensor variational inequalities. We investigate under what suitable assumptions the existence and, then, the uniqueness of solutions to generalized tensor variational inequalities are guaranteed. Moreover, we investigate on the approximation of solutions to the tensor variational inequality by using suitable perturbed tensor variational inequalities. We establish convergence of solutions to the regularized tensor variational inequalities to a solution of the original tensor variational inequality making use of the set convergence in Kuratowski’s sense (see [2]). After that, we focus our attention on some stability results. The introduction of such new class of variational inequalities is motivated to an important application to an economic model. More precisely, they express the generalization Cournot-Nash equilibrium conditions of a general oligopolistic market equilibrium model in which every firm produces several commodities. Thanks the variational formulation, it is possible to obtain the existence and the uniqueness of equilibrium distributions. At last we present the policy maker control equilibrium model (see [3]).

References

[1] A. Barbagallo, S. Guarino Lo Bianco, “Variational inequalities on a class of structured tensors”, accepted in J. Nonlinear Conv. Anal. 19 (2018) 711—729.

[2] A. Barbagallo, S. Guarino Lo Bianco, “On ill-posedness and stability of tensor variational inequalities: application to an economic equilibrium”, J. Global Optim. 77 (2020) 125–141.

[3] F. Anceschi, A. Barbagallo, S. Guarino Lo Bianco, “Inverse variational inequalities on a class of structured tensors”, preprint.

We prove functional inequalities on vector fields on the Euclidean space when it is equipped with a bounded measure that satisfies a Poincaré inequality, and study associated self-adjoint operators. The weighted Korn inequality compares the differential matrix, once projected orthogonally to certain finite-dimensional spaces, with its symmetric part and, in an improved form of the inequality, an additional term. We also consider Poincaré-Korn inequalities for estimating a projection of the vector field by the symmetric part of the differential matrix and zeroth-order versions of these inequalities obtained using the Witten-Laplace operator. The constants depend on geometric properties of the potential and the estimates are quantitative and constructive. These inequalities are motivated by kinetic theory and related with the Korn inequality (1906) in mechanics, on a bounded domain. Joint work with Kleber Carrapatoso, Jean Dolbeault, Frédéric Hérau and Stéphane Mischler.